S. L. Lee

Wavelets in wandering subspaces

Mallats construction, via a multiresolution approximation, of orthonormal wavelets generated by a single function is extended to wavelets generated by a finite set of functions. The connection between multiresolution approximation and the concept of wandering subspaces of unitary operators in Hilbert space is exploited in the general setting. An example of multiresolution approximation generated by cardinal Hermite B-splines is constructed

An algorithm for matrix extension and wavelet construction

This paper gives a practical method of extending an n x r matrix P(z), r ≤ n, with Laurent polynomial entries in one complex variable z, to a square matrix also with Laurent polynomial entries. If P(z) has orthonormal columns when z is restricted to the torus T, it can be extended to a paraunitary matrix. If P(z) has rank r for each z ∈ T, it can be extended to a matrix with nonvanishing determinant on T. The method is easily implemented in the computer. It is applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter.

Characterization of compactly supported refinable splines

AbstractWe prove that a compactly supported spline functionφ of degree k satisfies the scaling equation

SPECTRAL PROPERTIES OF CONTINUOUS REFINEMENT OPERATORS

\phi (x) = \sum _{n = 0}^N c(n)\phi (mx - n)

Wavelet bases for a set of commuting unitary operators

for some integerm ≥ 2, if and only if

Spectral properties of matrix continuous refinement operators

\phi (x) = \sum _n p(n)B_k (x - n)